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#help-23
safe radishBOT
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whole rivet
safe radishBOT
whole rivet
Hi! This is the problem and it being worked out all the way, its calculus related rates.
I understand the process, but here dx/dt is 1/6??
I would assume from the word problem clue that it'd be 2 inch/sec instead and not 1/6 inch/sec???
Unless the key is wrong
mossy lotus
2in = 1/6 ft
whole rivet
oh, I see, does it have to be converted?
mossy lotus
yea, like they said in the second line, they are working with the unit ft/sec
safe radishBOT
solar hazel
inches and feet unironically in a math problem
honest perch
lmao
safe radishBOT
Available help channel!
rotund ermine
I am writing the equation for a sinusoidal graph and basically I am wondering, can there be more than 1 right answer? Or what did I do wrong?
Because the graphs when on the calculator are different
safe radishBOT
I am writing the equation for a sinusoidal graph and basically I am wondering, c...
split kayak
I am writing the equation for a sinusoidal graph and basically I am wondering, c...
Your solution is off and i suspect it has something to do with parenthesis.
Althought, your idea of there being multiple solutions is true
astral glacier
Would be helpful if you walked us through how you arrived at your answer
That way we can pinpoint where you went wrong
rotund ermine
I found the amplitude & the midline by counting on the graph. and then for the period i subtracted the two x-values of the minimums.
then to figure out the phase shift i looked at where the first like "cos" graph would start which was at (-1/6) so then for the phase shift i subtracted that amount?
but like phase shift kind of confuses me 😭😭
astral glacier
Actually hold on
rotund ermine
bro wait
astral glacier
Hm?
rotund ermine
im in degree mode on mt calculator AGHH 😭😭😭
nvm
its the same graph in radian mode
SORRY 😭😭😭
astral glacier
Firstly, your answer is missing a parentheses somewhere
rotund ermine
oh
astral glacier
,w plot cos (2(x-pi/6)) + 1
flat frigateBOT
rotund ermine
wait
astral glacier
Unless you're doing some chicanery with parentheses
flat frigateBOT
flat frigateBOT
astral glacier
Yeah they're not the same graph cool
astral glacier
oh if i change the B to be (5/6) that would work?
What exactly is B
astral glacier
,w plot cos(2(x-5pi/6)) + cos(2(x-pi/3))
flat frigateBOT
rotund ermine
i remember my teacher said something about how you look to the right when writing sinusoids? if that makes sense. but i forgot that. but also maybe there's like a specific reason for that but i wouldn't know
austere goblet
hint: period of cos
safe radishBOT
@rotund ermine Has your question been resolved?
rotund ermine
pi?
blazing cloak
safe radishBOT
blazing cloak
How do i do 10
2 rev per min is 720 degrees per minute
Is thst useful
.close
safe radishBOT
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loud ember
When real number a,b,c [a,b,c] == a * (b * b - c * c)
Which one is 인수 of [a,b,c] + [b,c,a] + [c,a,b]?
safe radishBOT
When real number a,b,c [a,b,c] == a * (b * b - c * c)
Which one is 인수 of [a,b,c]...
main mural
what is 인수? google translate says it's "factor" but it doesn't make much sense with real numbers
peak estuary
are you asking what the factorization of a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2) is?
if it exists?
loud ember
It should find which term can make ([a,b,c] + [b,c,a] + [c,a,b] ) % (Term) == 0
peak estuary
you can imagine that a is a variable and b,c are constants
can you then find some values that if you plug them into a, the expression is zero?
loud ember
It should be guarenteed regardless of whichever a is so it might not work
peak estuary
do you know that to get such a term for example for x^2-5x+6 you would find a root and then divide out (x-root)?
its the same thing here
just different letters
the root would be in terms of b and c
loud ember
can you then find some values that if you plug them into a, the expression is ze...
None because it cannot cancel all terms containing b or c
peak estuary
what happens if you set a=b?
safe radishBOT
@loud ember Has your question been resolved?
safe radishBOT
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blazing smelt
safe radishBOT
open wedge
I think you can make the diagran yourself?
safe radishBOT
@blazing smelt Has your question been resolved?
blazing smelt
I think you can make the diagran yourself?
No i used geogebra too
I wasnt able to make one
I dont see the point where SAE and TCF meet
normal moss
GeoGebra 💪
safe radishBOT
@blazing smelt Has your question been resolved?
tardy mango
blazing smelt
https://www.geogebra.org/calculator/nxbfdafu
Thankyou sir
I didn't realise that I can twist the things to get the intersection
.close
safe radishBOT
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stiff vine
safe radishBOT
stiff vine
i do cross product and thats it?
median vigil
you mean take the curl?
median vigil
that would be sufficient to determine if it is conservative or not
stiff vine
that would be sufficient to determine if it is conservative or not
ok so what do the other 2 mean
median vigil
well it asks you to do something based on whether it's conservative or not
you find out which one once you have determined that
stiff vine
is potential function the one where u do integral of first thing, then do partial deirvative with respect to y and then set equal to the y in original equation
median vigil
that's the procedure for finding it
the idea is that it is a function whose gradient is that vector field
stiff vine
what about finding a loop whats the equation
median vigil
there isn't a specific equation, it could be any closed curve
you could give a parameterization
stiff vine
bruh whats parameterization even mean theres so much of them
median vigil
it's a function which gives all the points on the curve as a function of a single variable called the parameter
for example x = cos(t), y = sin(t), z = 0 for 0 <= t <= 2pi is a parameterization of the unit circle in the xy-plane
stiff vine
well it asks you to do something based on whether it's conservative or not
ok its conserative so for potential function i do integral of the stuff inside i first with respect to x?
median vigil
yes
desert pasture
is this baxendall
median vigil
you could integrate in any order but that's an option
stiff vine
is this baxendall
what
stiff vine
no its from a practice exam
safe radishBOT
@stiff vine Has your question been resolved?
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worthy badger
safe radishBOT
round egret
what makes you think it's -5 btw?
worthy badger
cuz when you put it in the claculator it gives you a -5
round egret
it's a common mistake so don't worry about it. Just be careful with the brackets
worthy badger
but how to close it
round egret
type in .close
queen slate
idk if u guys can read this but i know the answer idk how to get there
fathom adder
You can't simplify like this
queen slate
like what
fathom adder
Let that x² + 2 on top
fathom adder
No
queen slate
oh oops
winged flare
yeah the cancellation is bad
queen slate
what do i do
winged flare
let it be
fathom adder
Ye
queen slate
alright
winged flare
and simplify the numerator by expansion
fathom adder
Quotient rule is correct else
queen slate
and simplify the numerator by expansion
as in combine whatever i can?
is that what that means
winged flare
yeah
open wedge
idk if u guys can read this but i know the answer idk how to get there
product rule
winged flare
to match the options
marsh gorge
idk if u guys can read this but i know the answer idk how to get there
U can simplify the equation first
U can write the numerator as x^2+2 -4
Then u get
open wedge
U can write the numerator as x^2+2 -4
i don't think that's really needed
marsh gorge
i don't think that's really needed
Previous context he said he’s not great with algebra
Maybe it could help build intuition
fathom adder
U can write the numerator as x^2+2 -4
-4 yes
open wedge
Previous context he said he’s not great with algebra
product rule is enough
open wedge
i have 4x^3 + 4x) - ( 4x^3 - 4x all over (x^2 +2)^2
eh?
marsh gorge
shouldi write it
Ye
queen slate
idk how bc if i do the top it turns into 0
oh
do i distribute the
negative
sign
i think so imma try
winged flare
yeah distribute the negative sign
to get $=\frac{(2x^3+4x) - 2x^3 + 4x}{(x^2+2)^2}$
queen slate
oh ya i got B
flat frigateBOT
queen slate
queen slate
yea
winged flare
b it is
queen slate
i have this 1 questions and i have zero idea how to do it
bc i forgot log rules
and e rules
i tried to do product rule
winged flare
and what did that give you
queen slate
it gave me 1 over e plus ln e ^2
queen slate
uh
winged flare
$\frac{d}{dx}(ln(x^2))$ is not just $\frac{1}{x^2}$
flat frigateBOT
modest oar
Is 4 correct answer??
winged flare
Do they mean ln(x²) or ln²(x) in this case ?
former
winged flare
Is 4 correct answer??
appreciate your effort
queen slate
answer key it says C
winged flare
but at least spoiler
queen slate
Is 4 correct answer??
yea
modest oar
answer key it says C
Exactly it correct
winged flare
ok to continue
for not simple functions like ln(x^2)
you need to differentiate once more
modest oar
idk how to get there tho
Use uv rule
queen slate
you need to differentiate once more
so it wouldnt just be ln over x^2
winged flare
$\frac{d}{dx}(ln(x^2)) = \frac{1}{x^2} \cdot \frac{d}{dx}(x^2)$
queen slate
i mean
flat frigateBOT
modest oar
First differentiate x and keep log x sq. As it is
The add differentiation of log x sq. Into x
winged flare
$\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$
flat frigateBOT
modest oar
@queen slate see my previous message
winged flare
<@713262968732778516> see my previous message
ok look
they applied the product rule
the part bothering them is differentiating ln(x^2)
just to save you from explaining once more
modest oar
@winged flare r u indian
queen slate
ok look
they applied the product rule
the part bothering them is differentiating...
so i do chain rule to the ln (x^2) right
queen slate
<@702550180154834984> r u indian
dawg
winged flare
so i do chain rule to the ln (x^2) right
yes
winged flare
<@702550180154834984> r u indian
off topic but sure
modest oar
off topic but sure
Ok I am also indian
modest oar
off topic but sure
No problem it's off topic I guess double is solved
queen slate
is this not how to do chain rule
modest oar
is this not how to do chain rule
It's wrong
winged flare
is this not how to do chain rule
first differentiate similar to how you would differentiate ln(x)
modest oar
is this not how to do chain rule
Do 1/x square into 2x
winged flare
giving you $\frac{1}{x^2}$
flat frigateBOT
queen slate
ohh
winged flare
something like $ln(?) = \frac{1}{?}$
flat frigateBOT
winged flare
just imagine it as some sort of x
queen slate
so its like this or still wrong
winged flare
so its like this or still wrong
this
modest oar
so its like this or still wrong
This is correct
winged flare
yes
queen slate
winged flare
go on
nimble vine
ye
safe radishBOT
safe radishBOT
Available help channel!
queen slate
i thought it was between A or D and i checked and i got it fully wrong
queen slate
its C and idk how
cloud hound
Forgot the power rule did we
mighty mango
i thought it was between A or D and i checked and i got it fully wrong
the chain rule states
derivative of a function inside of another function : 'derivative of the outside * derivative of the inside"
queen slate
Forgot the power rule did we
i didnt know i have to use that here
cloud hound
Looks like car is explaining why
mighty mango
do you know chain rule?
queen slate
f(x) is sin^3 and g(x) is 4x
mighty mango
so... go ahead!
queen slate
i tried
mighty mango
retry.
mighty mango
i thought it was between A or D and i checked and i got it fully wrong
because the derivative of sin^3(x) is not cos^3(x)
queen slate
i thought it was between A or D and i checked and i got it fully wrong
to the side is my attempt at chain rule
queen slate
because the derivative of sin^3(x) is not cos^3(x)
is it 3 cos^2
how do i know when to use product rule and also chain rule both
bc i see i have to use product rule
mighty mango
because the derivative of sin^3(x) is not cos^3(x)
only the derivative of sin(x) is cos(x). otherwise, if the function is raised to a power, we treat what's "inside" of the power as a function.
sin^3(x) is just (sin(x) )^3 right? so how would you go on about differentiating (2x+1)^2
faint hornet
use product rule when you have multiple 'different' functions multiplied by each other
queen slate
and also the derivative thing with exponents also applies to trig functions? like sin^3 will be 3 sin ^2?
queen slate
use product rule when you have multiple 'different' functions multiplied by each...
alright
mighty mango
differentiate (2x+1)^2
queen slate
2(2x+1) times 2
mighty mango
you dont need to expand whatever brackets you'll come up with.
mighty mango
2(2x+1) times 2
perfect!
so what you did there was
you derived (2x+1)^2 as if it was x^2 and just used power rule. which gave you 2(2x+1) but then you mutliplied by the derivative of what's inside the bracket (aka 2x+1) which gave you 2
so lets try to apply that on sin^2(x)
quasi bison
you derived (2x+1)^2 as if it was x^2 and just used power rule. which gave you 2...
differentiated :^)
not derived
mighty mango
it is just ( sin(x) )^2
queen slate
so lets try to apply that on sin^2(x)
so the derivative of sin^2(x) is ( sin(x) )^2 ??
quasi bison
no, sin^2(x) itself is (sin(x))^2
queen slate
oh okok
mighty mango
differentiated :^)
its too much to write ann 😔
this is exactly what you did in the (2x+1)^2 situation
used power rule on (2x+1)^2 then multiplied that by the derivative of 2x+1
queen slate
ok so sin stays sin but 3 sin ^2 times cos?
mighty mango
ok so sin stays sin but 3 sin ^2 times cos?
you might be onto something but i dont quite understand
so perhaps type what you wanna do out
mighty mango
you need to ignore your original problem
I'm just asking you to try and differentiate ( sin(x) )^2
queen slate
2 sin (x) times cos?
mighty mango
if you cant, I'll just help you. so ask!
winged flare
i thought it was between A or D and i checked and i got it fully wrong
clearer notation would look like\
$f(x) = (sin(4x))^3$
flat frigateBOT
mighty mango
2 sin (x) times cos?
CORRECT!
perfect
you used power rule on the whole thing then multiplied by the derivative of the inside
queen slate
yea
mighty mango
that's exactly what we're gonna do with your original problem
queen slate
3 sin^2x 4x times cos (4)
idk
mighty mango
the first part was very right
by using power rule
3 sin^2x (4x)
wait what
you need brackets...
open wedge
$f(x) = (\sin 4x)^3$
flat frigateBOT
1 divided by 0 equals Infinity
open wedge
3sin^2(4x) with power rule
wait HUH?
open wedge
chain rule: $h(x) = f(g(x)) \Rightarrow h'(x) = f'(g(x)) \cdot g'(x)$
flat frigateBOT
1 divided by 0 equals Infinity
open wedge
mighty mango
now onto the next step
.
mighty mango
which is derivative of the inside
.
mighty mango
3sin^2(4x) with power rule
that is not hte final answer.
mighty mango
chain rule: $h(x) = f(g(x)) \Rightarrow h'(x) = f'(g(x)) \cdot g'(x)$
if... that's what you're concerned with?
queen slate
what is the inside?
cos 4?
mighty mango
cos 4?
I was lowkey asking for the name of the inside and not hte derivative of it
so yeah the inside is sin(4x)
how would you differentiate that
queen slate
cos (4x) times 4??\
mighty mango
exactly!!!
so
derivative of the outside with power rule is 3sin^2(4x)
derivative of the inside is 4cos(4x)
you just multiply them!
which gives you c!!!!!!!
yaaaaay
notice 3 * 4 = 12
so
12sin^2(4x)cos(4x)
would you perhaps want another exercise with trig functions that need chain rule?
and no, you didnt need product rule. you could, but you dont wanna product rule 3 functions.
mighty mango
use product rule when you have multiple 'different' functions multiplied by each...
as stated by wojciech, it is only preferred to use product rule when you have multiple different functions.
faint hornet
and no, you didnt need product rule. you could, but you dont wanna product rule ...
this is not entirely true fwiw
it is perfectly normal and okay to use it on 3 functions, but there is no point when there are more efficient methods
mighty mango
sure! always assume there are other efficient methods than power rule'ing three functions espiecally with trig! soon you'll be needing trig identities and you'll def use them!
@queen slate are you still there?
I wouldn't differentiate 2sin(x)cos(x)csc(2x) with product even if i ran out of every idea 😔.
actually, OP, would you like to try to differentiate that?
if... you're even here
safe radishBOT
@queen slate Has your question been resolved?
south sparrow
Why is this profound and different, what is the big idea? In the equation which moves to notice?
safe radishBOT
Why is this profound and different, what is the big idea? In the equation which ...
south sparrow
I don't yet fully appreciate / understand why this is nice
It's from Differential Geometry and Form by Tristan Needham
feral linden
It gives you a intuitive explanation of tan’(t)=1/cos^2(t)
safe radishBOT
@south sparrow Has your question been resolved?
south sparrow
Thank you @feral linden
safe radishBOT
south sparrow
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Available help channel!
summer sphinx
When I solved it I got 84 but the total combinations possible 9C3 is also 84 so I probably made a mistake but I don’t see anything wrong with what I did
safe radishBOT
When I solved it I got 84 but the total combinations possible 9C3 is also 84 so ...
cloud hound
You were almost there but 9C3 is just the number of different ways we can take 3 balls from 9, not necessarily that the black ball has to be included
summer sphinx
So I did 3C1 to get the black balls and then I multiplied it by 8C2
For the remaining 2 balls
cloud hound
So I did 3C1 to get the black balls and then I multiplied it by 8C2
There are a few ways to do this
I would say the easiest is to continue in this way
9C3 is all the ways we can pull 3 balls, let's subtract the number of ways we can pull only non black balls
That will give us the number of ways to pull at least one
summer sphinx
I get that, but why doesn’t my method work
solar hazel
because you can get black balls in ‘the 8’ which leads to you counting things more than once
cloud hound
Because it counts the number of ways to get 1 black ball and two balls from the others, double counting when you get another black ball
What slayla said
solar hazel
what coolempire said
summer sphinx
But in the question they said at least 1 black ball, not 1 black ball. At least
solar hazel
imagine the times that you pick a black ball in the other 8
that event would be counted again when you pick that one as “the first” black ball
primal bone
But in the question they said at least 1 black ball, not 1 black ball. At least
On this note, a more efficient approach would be to count how many ways you can get no black balls, then subtract it from the total number of configurations
primal bone
merde
-# shit
summer sphinx
I know that, but I didn’t get why my method was incorrect
Welp, I’m gonna close this now
cloud hound
merde
-# shit
Le sous-titre me tue.
-# The subtitle is killing me
summer sphinx
.close
safe radishBOT
cloud hound
Oh I didn't get to go over the other cool ways to do it 😆 which is counting 1 black ball + 2 black balls + 3 black balls
My preferred method tbh
summer sphinx
Welp, that’s how my friends did it
cloud hound
,w (3 choose 1)(6 choose 2) + (3 choose 2)(6 choose 1) + (3 choose 3)
flat frigateBOT
cloud hound
Correct!
nimble vine
all - none requires less work tbh
cloud hound
It is less work to do it that way true
I always do my 'at least' this way for some reason but maybe I'll change
safe radishBOT
Available help channel!
thin lion
Hi is this proof valid?
safe radishBOT
Hi is this proof valid?
proud tree
Hi is this proof valid?
seems about right
safe radishBOT
thin lion
.reopen
safe radishBOT
✅ Original question: #help-23 message
thin lion
could i please have a small hint for this
im attempting to do the scracthwork
so i choose an epsilon
idk how to begin
spiral bronze
multiply by the conjugate over itself?
cloud hound
That's what I was going to say as well
thin lion
is that a usual trick?
spiral bronze
Yes
cloud hound
When you have a sub/difference of square roots yeah
thin lion
oh i see, ill try it ty
spiral bronze
Good luck!
spiral bronze
Have you done the sandwich theorem?
spiral bronze
Well what's a good lower bound?
thin lion
-1/sqrt(n) ?
spiral bronze
I'd say a constant, but that works to tbh
thin lion
0 then in that case?
spiral bronze
That's what I would do, but what you said works too :)
So long as the sandwich theorem holds, you're done
spiral bronze
Okie dokie
cloud hound
I like the variable one as well 🙂
spiral bronze
I like the variable one as well 🙂
My thought was that it's relatively easy to justify that the result is always positive
thin lion
I think I need to be clear on the steps, idek how to write it formally 😭
Like in a logical order
thin lion
My thought was that it's relatively easy to justify that the result is always po...
Hm we know both n and n+1 are natural numbers so greater than 0 so is it justified enough to say that the reciprocal of them is also greater than 0?
spiral bronze
Hm we know both n and n+1 are natural numbers so greater than 0 so is it justifi...
It's correct, but it depends upon what you did in your lectures
cloud hound
Like in a logical order
- choose epsilon
- plug epsilon in where it's supposed to go
- essentially do the exact reverse of the steps in your scratch work
- proof complete
is usually how it goes
thin lion
Wait in our case what exactly would eps be 😭 cause we used sandwich right, would it just be 1/sqrt n?
But then the lower bound?
cloud hound
Wait in our case what exactly would eps be 😭 cause we used sandwich right, woul...
Yeah
cloud hound
But then the lower bound?
What exactly happens when you remove an absolute value sign
sick plaza
The equation x^2 + 2x = k(x-1) has non zero roots where the difference between the roots is 2 , find the value of each root and the value of k.
safe radishBOT
The equation x^2 + 2x = k(x-1) has non zero roots where the difference between t...
normal moss
!status
safe radishBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
uhm
i knew x^2 - (sum of roots)x + product of roots =0
but im not sure of the roots
like it says the difference of the roots is 2, but im not sure should i write a+2? a-2? a+-2? or y
nimble vine
like it says the difference of the roots is 2, but im not sure should i write a+...
what is a
nimble vine
a-b = 2
sick plaza
and i did x^2 +2x-kx+k=0 because i know i have to compare and find the value
woeful stag
Turn it into a standard form and use Vieta's formula
normal moss
and i did x^2 +2x-kx+k=0 because i know i have to compare and find the value
What is the difference of roots
sick plaza
2
normal moss
Well, yes
woeful stag
or..
normal moss
In a quadratic in the form of $ax^2 + bx + c = 0$
flat frigateBOT
USS-Enterprise
normal moss
You get a nice formula for the difference of roots $x_1$ and $x_2$, or $\alpha$ and $\beta$ if you will
flat frigateBOT
USS-Enterprise
sick plaza
if i will ?
nimble vine
like it says the difference of the roots is 2, but im not sure should i write a+...
if you want to go with this then a-2 and a
normal moss
so what should i do next
Do you know the formula?
sick plaza
which one?
normal moss
For the difference of roots
sick plaza
x^2 - (Sum of roots)x + Product of roots =0?
normal moss
x^2 - (Sum of roots)x + Product of roots =0?
It's sum of roots, not sum of products
normal moss
But this really just beats around the bush
Ok
So how do we get the first root
$\alpha$
flat frigateBOT
USS-Enterprise
normal moss
If our quadratic is in the form of $ax^2 + bx + c = 0$
flat frigateBOT
USS-Enterprise
normal moss
That would be the solution to a linear equation, if we had ax + b = 0
Think of the quadratic formula
sick plaza
-b+- sqrt b^2-4ac/2a?
normal moss
Yes
For the first root, we take + square root
For the second, -
So we get
$\alpha = \frac{-b + \sqrt{b^2-4ac}}{2a}$
flat frigateBOT
USS-Enterprise
flat frigateBOT
USS-Enterprise
normal moss
Correct?
sick plaza
yea
flat frigateBOT
USS-Enterprise
normal moss
So what would $\alpha - \beta$ be
flat frigateBOT
USS-Enterprise
sick plaza
2?
normal moss
$\alpha = \frac{-b + \sqrt{b^2-4ac}}{2a}$
But you have alpha
And you have beta
So do alpha - beta
sick plaza
sqrt b^2 -4ac + sqrt -b^2 +4ac?
normal moss
$\alpha = \frac{-b + \sqrt{b^2-4ac}}{2a}$
flat frigateBOT
USS-Enterprise
normal moss
$\beta = \frac{-b - \sqrt{b^2-4ac}}{2a}$
flat frigateBOT
USS-Enterprise
normal moss
Therefore, $\alpha - \beta = \frac{-b + \sqrt{b^2-4ac}}{2a} - \frac{-b - \sqrt{b^2-4ac}}{2a}$
flat frigateBOT
USS-Enterprise
sick plaza
yea
normal moss
Yes
flat frigateBOT
USS-Enterprise
sick plaza
-1
sick plaza
oh
normal moss
Nothing inside the square root gets affected
normal moss
And what is that
sick plaza
then sqrt b^2-4ac/a
flat frigateBOT
USS-Enterprise
normal moss
Now
normal moss
and i did x^2 +2x-kx+k=0 because i know i have to compare and find the value
You got this
$x^2 + 2x - kx + k = 0$
flat frigateBOT
USS-Enterprise
normal moss
We need it in the form of $ax^2 + bx + c = 0$ right
flat frigateBOT
USS-Enterprise
normal moss
So what can we do
sick plaza
square both side?
flat frigateBOT
USS-Enterprise
sick plaza
1
normal moss
If we want $ax^2 + bx + c = 0$
flat frigateBOT
USS-Enterprise
normal moss
Exactly
Then the second term is b times x
So how do we get 2x - kx to be b times x
sick plaza
2x-kx=bx?
normal moss
Sure
sick plaza
2-k=b?
normal moss
So what is b
Exactly
We factor out x
from 2x - kx
To get (2-k)*x
And b = 2 - k
So we have transformed the equation to
$x^2 + (2-k)x + k = 0$
flat frigateBOT
USS-Enterprise
normal moss
Everything clear so far?
sick plaza
yea
normal moss
a = 1; b = 2 - k; c = k
sick plaza
but i can never think of these in exam
normal moss
I mean.
All we did is expressed alpha and beta using the quadratic formula
Then literally just calculated alpha - beta
And then we rewrote the original equation into the form of ax^2 + bx + c = 0
That's it
You just factor out x from all terms which have x
that's all
But anyway
Now we get back to this
$\alpha - \beta = \frac{\sqrt{b^2 - 4ac}}{a}$
flat frigateBOT
USS-Enterprise
normal moss
We know alpha - beta = 2, right
So we can rewrite it as
$2 = \frac{\sqrt{b^2 - 4ac}}{a}$
flat frigateBOT
USS-Enterprise
flat frigateBOT
USS-Enterprise
normal moss
Following?
sick plaza
yea
sick plaza
1
flat frigateBOT
USS-Enterprise
normal moss
Right?
sick plaza
yea
normal moss
$x^2 + (2-k)x + k = 0$
From here
sick plaza
b is 2-k
normal moss
Yes
sick plaza
c is k?
flat frigateBOT
USS-Enterprise
sick plaza
k=8?
normal moss
**USS-Enterprise**
But to this you get two k's
So
We substitute in
We get
$\sqrt{(2-k)^2 - 4 \cdot k} = 2$
flat frigateBOT
USS-Enterprise
normal moss
Right?
sick plaza
yea
normal moss
What did you do then
normal moss
Yes.
sick plaza
so k=0 or k=8?
normal moss
Exactly
And now we go back to our original equation
$x^2 + (2-k)x + k = 0$
And we have to test both k = 0 and k = 8
flat frigateBOT
USS-Enterprise
sick plaza
sub k=8 into it?
normal moss
I am very sorry. I am on a train, and in nearly 2026 we have extremely poor mobile data coverage on the train line between the country's two biggest cities 🤦
normal moss
sub k=8 into it?
Yes
nimble vine
I am very sorry. I am on a train, and in nearly 2026 we have extremely poor mobi...
what country
sick plaza
I am very sorry. I am on a train, and in nearly 2026 we have extremely poor mobi...
its okay
normal moss
The equation x^2 + 2x = k(x-1) has non zero roots where the difference between t...
And we need to see if we get 2 nonzero roots
normal moss
what country
Slovenia
normal moss
I mean, over 2.5 hours to travel 150km. Not to mention regular 20+min delays
A disgrace
normal moss
very cold country 🥶
Oh I find it very hot especially in the summer
It's fine in the winter
normal moss
I like cold more
normal moss
And we need to see if we get 2 nonzero roots
Anyway this is the point
We sub in k = 0, and k = 8
For both equations we find the roots
sick plaza
I like cold more
me too
normal moss
And to be considered a solution, it must have 2 nonzero solutions
sick plaza
yea
normal moss
So what'd you get 🙂
sick plaza
if i sub in k=8 ?
normal moss
For both
sick plaza
x^2+2x =0(x-1)
x^2+2x=8(x-1)
sick plaza
x^2+2x=0
x^2+2x=8x-8
normal moss
Yes
sick plaza
x^2-6x+8=0
normal moss
Yeah
nimble vine
x^2+2x =0(x-1)
x^2+2x=8(x-1)
if k= 0 one of roots become 0
they ask for non zero roots
nimble vine
you have k
normal moss
Well
You need to say that x = 0 is a solution to the first equation (when k = 0)
Therefore it doesn't have only nonzero solutions
So k = 0 is not a solution
sick plaza
ohh
so i write x is 0 when k is 0, x cant be non zero roots, therefore k cant be equal to 0?
normal moss
You write it like this
We have two options: k = 0, k = 8
When k = 0, we get $x^2 + 2x = 0$
flat frigateBOT
USS-Enterprise
normal moss
Factoring, we get $x \cdot (x + 2) = 0$
flat frigateBOT
USS-Enterprise
normal moss
This means the solutions to this equation are $x_1 = 0$ and $x_2 = -2$
flat frigateBOT
USS-Enterprise
normal moss
But we are only looking for nonzero solutions
0 is not nonzero, therefore k = 0 is not a solution.
And then you do the same thing for k = 8:
And you get solutions $x_1 = 2$ and $x_2 = 4$
flat frigateBOT
USS-Enterprise
normal moss
Which are both nonzero
sick plaza
can i just write no solution under the x1=0?
normal moss
Therefore the solutions are k = 8, x1 = 2, x2 = 4
normal moss
can i just write no solution under the x1=0?
No, that is incorrect
sick plaza
eh wait i cant
normal moss
As x = 0 is a solution
sick plaza
0 is not nonzero, therefore k = 0 is not a solution.
so i just write this word into it?
sick plaza
ohh
normal moss
Just write x = 0 is not a nonzero solution
sick plaza
ohhhh
normal moss
And therefore k = 0 is not a solution
sick plaza
imma take some time to digest it
normal moss
👍
sick plaza
thankss btw
normal moss
Hope I've been of help 🙂
sick plaza
of course you've
sick plaza
yea i will soon after i digest eerything
normal moss
i swear an hour goes by so quick
sick plaza
when can i use this like the quadratic formula
flat frigateBOT
USS-Enterprise
normal moss
Is used to find solutions to the equation in the form of $ax^2 + bx + c = 0$
flat frigateBOT
USS-Enterprise
normal moss
Where $x_1$ and $x_2$ are the solutions
flat frigateBOT
USS-Enterprise
normal moss
solving maths takes lots of time
It does
normal moss
so when it is given 2 roots, u can apply this to find?
It really depends on what your task is
Here it was useful, yes
Because we were given the difference of the roots
sick plaza
oh
normal moss
And we can calculate the difference very easily just by doing x_1 - x_2
But it really depends on the example
sick plaza
ohh
normal moss
Well anyway
I am about to arrive at the station
So I gotta go
But yeah, again, good luck with everything 🙂
See you around
verbal cloud
Define the sequence of real valued functions $h_n$ by $h_0=1$ and for all $n \in \mathbb{N}$ ,
$$h_{n+1} : x \mapsto h_n(\frac{x}{2}) +\frac{x}{2} \left(h_n\left(\frac{x}{2}\right) \right) ^2$$
Show that $h_n$ converges uniformly on $[0; 1]$
safe radishBOT
Define the sequence of real valued functions $h_n$ by $h_0=1$ and for all $n \in...
flat frigateBOT
bloubbloub
obsidian oracle
Define the sequence of real valued functions $h_n$ by $h_0=1$ and for all $n \in...
!status
safe radishBOT
Define the sequence of real valued functions $h_n$ by $h_0=1$ and for all $n \in...
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
@verbal cloud Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
blissful tapir
For what Smallest possible odd value of $n$ is $^n C _2$ divisible by 2024?
flat frigateBOT
Ishmam
brave wolf
For what Smallest possible odd value of $n$ is $^n C _2$ divisible by 2024?
do u know the formula for nC2?
blissful tapir
Yeah
brave wolf
cool, what is it?
blissful tapir
$\frac{n(n-1)}{2}$
flat frigateBOT
Ishmam
brave wolf
now I'd try prime-factorizing 2024
so that you can check divisibility by each prime factor instead
blissful tapir
2^3 * 11 * 23
brave wolf
okay, actually, lets rearrange it to
n(n-1) is divisible by 4048
so we will just add one 2
blissful tapir
Okay
brave wolf
2^4 * 11 * 23
blissful tapir
Okay
brave wolf
only one of n, n-1 can be divisible by 2
so that one will have to be divisible by 2^4
and n must be odd
blissful tapir
I think since n is odd, n-1 has to do it
brave wolf
alright, so n-1 must be divisible by 16
blissful tapir
True
brave wolf
now we will gotta do some casework i guess
brave wolf
now i guess we would have to do some casework
blissful tapir
I just don't wanna do casework 
brave wolf
either n is divisible by both 23 and 11, or only by 23 or only by 11 (or by neither, but that'd make n very high)
its just a few cases, so it shouldnt be that bad
,calc 23*11
flat frigateBOT
Result:
253
if its divisible by both, then we are looking for k such that
253k = 1 (mod 16)
this shouldnt be too bad
blissful tapir
Okay actually yeah
brave wolf
253 can be simplified, and after that u can either find the inverse, or quicker, bruteforce small k until u find a sol
blissful tapir
if its divisible by both, then we are looking for k such that
253k = 1 (mod 16)
k = 3 works there
flat frigateBOT
Result:
13
,calc 13*3 mod 16
flat frigateBOT
Result:
7
doesnt look right
blissful tapir
,calc 253 * 3 mod 16
flat frigateBOT
Result:
7
u can replace 253 with 13
since 253 = 13 (mod 16)
so just check multiples of 13
u can even skip the even ones (as theyll be even mod 16)
flat frigateBOT
Result:
1
Omg
flat frigateBOT
Result:
1265
this is probably one of the higher estimates
meager cedar
brave wolf
what if n-1 was divisible by 16 and 11 and n by 23
blissful tapir
That would be crazy big
blissful tapir
The problem is supposed to be solvable with only pen and paper
flat frigateBOT
Result:
176
Maybe try euclidean algorithm on pairs (11 13, 16), (11, 13 16), (13, 11 16) to find one solution? (A solution (x,y) of ax+by=1 gives you solutions (a-tb, b+ta): t
brave wolf
176k+1 must be divisible by 23
i.e. 176k = -1 mod 23 (we will be solving for n-1 now)
brave wolf
Maybe try euclidean algorithm on pairs (11 13, 16), (11, 13 16), (13, 11 16) to ...
these arent pairs though
oh is it supposed to be *
feral linden
I mean do euclidean algorithm on gcd(11 times 13, 16)=1 for example
brave wolf
then that's essentially what we are doing, except we are doing it in brute-force way, because i think that it's slightly quicker
brave wolf
176k+1 must be divisible by 23
i.e. 176k = -1 mod 23 (we will be solving for n-1...
this is still solvable with pen and paper
but yeah, euclidean algo could surely be used (afterall, what we are doing is essentially just finding the modular inverse, solving simple linear modular congruences)
brave wolf
176k+1 must be divisible by 23
i.e. 176k = -1 mod 23 (we will be solving for n-1...
||Reduce mod 23, we get 15k = -1 (mod 23). Trying first few k, we see that k = 3 already works. Done. It's really not that difficult to do it by hand. Note that we wouldnt have to go higher than k=7 anyway, since after that the sol would be higher than our previously found one||
blissful tapir
What about the minimal n requirement
brave wolf
idk how long is it supposed to take, but it should be doable in under 10 mins
brave wolf
What about the minimal n requirement
thats why we are trying all 3 cases
and we try k from lowest to higher
in the end, we choose the lowest solution we found
for the first case, the lowest solution was n=1265
flat frigateBOT
Result:
529
This is lower
safe radishBOT
brave wolf
Alright thanks
btw notice that when you will be trying the k for the third case, you dont even have to actually find it
,calc 23*16
flat frigateBOT
Result:
368
368*k will be higher than 529 for all k other than k = 1
so you only need to try k = 1
safe radishBOT
Available help channel!
eternal breach
is there any french speaking person that can help me do question 3)c
cursive remnant
is there any french speaking person that can help me do question 3)c
You have to determine if the sequence u_n converges, and if so, determine what it converges to
eternal breach
You have to determine if the sequence u_n converges, and if so, determine what i...
yes I need to deduce if Un is decreasing or increasing using the previous question but how
cursive remnant
yes I need to deduce if Un is decreasing or increasing using the previous questi...
do you want a hint or more involved help?
onyx valley
is there any french speaking person that can help me do question 3)c
comparaison logarithmique
je pense
safe radishBOT
@eternal breach Has your question been resolved?
onyx valley
en tout cas pour la limite c'est clairement de la comparaison logarithmique
et pour montrer la convergence ... je reflechis
ouais non enft tu fais les 2 en meme temps je pense
verbal cloud
Define the sequence of real valued functions $h_n$ by $h0=1$ and for all $n \in \mathbb{N}$ ,
$$h_{n+1} : x \mapsto h_n(\frac{x}{2}) -\frac{x}{2} \left(h_n\left(\frac{x}{2}\right) \right) ^2$$
Show that $h_n$ converges uniformly on $[0; 1]$
safe radishBOT
Define the sequence of real valued functions $h_n$ by $h0=1$ and for all $n \in ...
verbal cloud
reopening this as I had to leave the other day
we're supposed to use the fact thas $T_x: y \mapsto y - \frac{xy^2}{2}$ is 1-lipschitz and $T_x([0; 1]) \subset [0; 1]$.
verbal cloud
but I was thinking maybe $h_{n+1} - h_n$ would be around $\frac{1}{2^n}$ but the fact that it's defined with $x/2$ is annoying
flat frigateBOT
bloubbloub
safe radishBOT
@verbal cloud Has your question been resolved?
feral linden
Define the sequence of real valued functions $h_n$ by $h0=1$ and for all $n \in ...
Not sure it helps, but by induction we can show that |h_n|<=1 and h_n are all 1-Lipschetz
feral linden
This is the way:
safe radishBOT
This is the way:
kind seal
. Lol
feral linden
Define the sequence of real valued functions $h_n$ by $h0=1$ and for all $n \in ...
1: |h_n|<=1 and all h_n are 1-Lipschitz by induction
2: Tx(y)<=y thus h_n(x) monotonically decreases
3: by 1 and 2 h_n(x) pointwisely converges to a 1-Lipschitz (thus continuous ) h
4: thus we have Dini’s theorem to apply
Though I am not familiar with the details of Dini’s theorem. I just know there is such a thing we can use
Anyway I probably should close it now
.close
verbal cloud
1: |h_n|<=1 and all h_n are 1-Lipschitz by induction
2: Tx(y)<=y thus h_n(x) mon...
I don't get point 2
safe radishBOT
I don't get point 2
feral linden
I don't get point 2
Oh shit… h_n+1(x)=Tx(h_n(x/2)) no wonder you said this x/2 is annoying…
Sorry, don’t have a clue yet…
long finch
Damn it I shouldn’t have joined something like this 😭 I’m fucking in the 6th grade
safe radishBOT
@verbal cloud Has your question been resolved?
lone void
arent 6th graders only 12 💀
random condor
Damn it I shouldn’t have joined something like this 😭 I’m fucking in the 6th gr...
One message removed from a suspended account.
iron estuary
can we show that $|h_{n+1}-h_n|_\infty \le 2^{-n}$ which would imply uniform convergence by the cauchy criterion?
flat frigateBOT
anflo
astral glacier
!redir @long finch @lone void @random condor
safe radishBOT
This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.
Available help channel!
hard sparrow
alright im an average 9th grader and I need help with constructions on numberlines and even normal constructions as I dont understand them for now, for example construction of 60 degrees, 120 and etc
safe radishBOT
alright im an average 9th grader and I need help with constructions on numberlin...
mossy lotus
well, 60deg is a pretty significant angle, coz its the internal angle of an equilateral triangle
so the way to construct is to essentially construct an equilateral triangle
You draw a random line and make a segment of finite length on it, and use your compass to draw the arcs of the radius same as that of the segments length
safe radishBOT
@hard sparrow Has your question been resolved?
hard sparrow
im ngl i dont understand
you jus tdraw random lines?
im sorry im bad at constructions
mossy lotus
to start with? yes
sometimes, it is a related to other parts of a figure, in which case you need to identify a correct line
mossy lotus
im ngl i dont understand
also you should react with a ❌ to the bot so it doesnt close the channel prematurely
hard sparrow
also you should react with a ❌ to the bot so it doesnt close the channel prematu...
oh
hard sparrow
sometimes, it is a related to other parts of a figure, in which case you need to...
can I send photos?
split fulcrum
baktep
hard sparrow
crap
hard sparrow
better be related to the topic tho
im rly sorry i gtg atm
could I make this ticket later again?
mossy lotus
yea, you can pick an open channel whenever